Henri Poincare and the Chaotic Whirlwind of Knowledge: on Performance of Mathematical Thinking

The importance of visualizations for mathematical and physical research in nonlinear dynamics that became known in the 1980ies under the buzzword of 'chaos theory' can hardly be doubted. This paper, however, focuses on the role of imagery in the work one of the uncontested founders of this field in the 19th century: Henri Poincare. Grappling with the complex interdependency of drawings, mental imagery and verbal metaphors, it argues that the caveats against images as well as the concurrent search for 'missing' images are crucial moments to approach and understand both Poincare's usage of imagery and their meaning for his own thinking and the invention of basic concepts in nonlinear science, i.e. the homoclinic point. Poincare's philosophical writings contain elements of a pictorial epistemology that are discussed with regard to his last unpublished paper. The essay concludes that nonlinear science already had been an 'image science' long before the age of digital computers.